FOR FREE MATERIALS

sigma (∑) is  a string alphabet  that consists of a finite set of symbols. 

Example ∑ = {a, b} or ∑ = {0,1} or ∑ = {a, b, c}

Power of an Alphabet Sigma (∑) in Automata, Kleene Closure in TOC, Positive closure:

Here ∑n   (for some integer n) denotes the set of strings of length n with symbols from sigma (∑). 

In other words,

∑n  = {w | w is a string over and | w | = n}. 

Hence, for any alphabet, ∑0   denotes the set of all strings of length zero. That is, = {e}. 

Let ∑ = {a, b} then

∑0   = {λ}

∑1   = {a, b}

∑2   = (a, b).(a ,b) = aa, ab, ba, bb (4 string with two length over a,b)

So,  ∑n   = 2n number of string = | ∑n  |

• At most n length: -

Kleene closure definition:

∑* :- The set of all strings over an alphabet is denoted by ∑*  . It contains set of all the strings that can be generated by iteratively concatenating symbols from any number of times. That is,

Kleene closure example:

Let ∑ = {a,b} then

∑*  is a set of all strings generated by a, b. Basically ∑*  is a universal set over a,b i.e. 

 (a + b)*.

Here ∑*   = { ε, a, b, aa, ab, ba, bb, aaa, aab, aba, abb, baa, …} 

[We can write ε or λ any one]

Positive closure:

∑+   :- The set of all strings over an alphabet excluding λ  is denoted by ∑+  . That is,

Example: 

Let ∑ = {a,b} then

∑+  is a set of all strings generated by a and b with at least one length of the string. 

∑+   is also universal set over a,b but excluding  i.e. (a + b)+

Here ∑+   = {a, b, aa, ab, ba, bb, aaa, aab, aba, abb, baa, …} 

Note :  To understand the concept of ∑*   and ∑+    is very important for Automata.